\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  -23.0 & -3.00 x_{1} & -7.00 x_{2} & +  7.00 x_{3} & + 10.00 x_{4} & + 10.00 x_{5} & -7.00 x_{6} & -3.00 x_{7}\\
 x_{9}   &  21.0 & -8.00 x_{1} & +  7.00 x_{2} & +  2.00 x_{3} & +  9.00 x_{4} & -1.00 x_{5} & -10.00 x_{6} & -9.00 x_{7}\\
 x_{10}   &  44.0 & -7.00 x_{1} & -8.00 x_{2} & +  4.00 x_{3} &   & -4.00 x_{5} & -9.00 x_{6} & + 10.00 x_{7}\\
 x_{11}   &  -5.0 & +  5.00 x_{1} & -5.00 x_{2} & -1.00 x_{3} & +  6.00 x_{4} & -6.00 x_{5} & +  3.00 x_{6} & -1.00 x_{7}\\
 x_{12}   &  -4.0 & +  3.00 x_{1} & -2.00 x_{2} & -9.00 x_{3} & +  3.00 x_{4} &   & -7.00 x_{6} & -1.00 x_{7}\\
 x_{13}   &  1.0 & +  9.00 x_{1} & +  4.00 x_{2} & +  4.00 x_{3} & +  5.00 x_{4} & +  1.00 x_{5} & -10.00 x_{6} & -5.00 x_{7}\\
 x_{14}   &  1.0 & +  9.00 x_{1} & -5.00 x_{2} & +  4.00 x_{3} & +  4.00 x_{4} & -7.00 x_{5} & +  7.00 x_{6} & +  9.00 x_{7}\\
 x_{15}   &  29.0 & -7.00 x_{1} & -3.00 x_{2} & +  9.00 x_{3} & +  7.00 x_{4} & -5.00 x_{5} & -6.00 x_{6} & -2.00 x_{7}\\
\hline
z    &  0.0 & +  1.00 x_{1} & +  3.00 x_{2} &   & +  1.00 x_{4} & -3.00 x_{5} &   & +  2.00 x_{7}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{7}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,15$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  3.00 y_{8} & +  8.00 y_{9} & +  7.00 y_{10} & -5.00 y_{11} & -3.00 y_{12} & -9.00 y_{13} & -9.00 y_{14} & +  7.00 y_{15}\\
 y_{2}   &  1.0 & +  7.00 y_{8} & -7.00 y_{9} & +  8.00 y_{10} & +  5.00 y_{11} & +  2.00 y_{12} & -4.00 y_{13} & +  5.00 y_{14} & +  3.00 y_{15}\\
 y_{3}   &  1.0 & -7.00 y_{8} & -2.00 y_{9} & -4.00 y_{10} & +  1.00 y_{11} & +  9.00 y_{12} & -4.00 y_{13} & -4.00 y_{14} & -9.00 y_{15}\\
 y_{4}   &  1.0 & -10.00 y_{8} & -9.00 y_{9} &   & -6.00 y_{11} & -3.00 y_{12} & -5.00 y_{13} & -4.00 y_{14} & -7.00 y_{15}\\
 y_{5}   &  1.0 & -10.00 y_{8} & +  1.00 y_{9} & +  4.00 y_{10} & +  6.00 y_{11} &   & -1.00 y_{13} & +  7.00 y_{14} & +  5.00 y_{15}\\
 y_{6}   &  1.0 & +  7.00 y_{8} & + 10.00 y_{9} & +  9.00 y_{10} & -3.00 y_{11} & +  7.00 y_{12} & + 10.00 y_{13} & -7.00 y_{14} & +  6.00 y_{15}\\
 y_{7}   &  1.0 & +  3.00 y_{8} & +  9.00 y_{9} & -10.00 y_{10} & +  1.00 y_{11} & +  1.00 y_{12} & +  5.00 y_{13} & -9.00 y_{14} & +  2.00 y_{15}\\
\hline
z    &  -0 & + 23.00 y_{8} & -21.00 y_{9} & -44.00 y_{10} & +  5.00 y_{11} & +  4.00 y_{12} & -1.00 y_{13} & -1.00 y_{14} & -29.00 y_{15}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 2 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.3 & -0.30 y_{4} & +  5.30 y_{9} & +  7.00 y_{10} & -6.80 y_{11} & -3.90 y_{12} & -10.50 y_{13} & -10.20 y_{14} & +  4.90 y_{15}\\
 y_{2}   &  1.7 & -0.70 y_{4} & -13.30 y_{9} & +  8.00 y_{10} & +  0.80 y_{11} & -0.10 y_{12} & -7.50 y_{13} & +  2.20 y_{14} & -1.90 y_{15}\\
 y_{3}   &  0.3 & +  0.70 y_{4} & +  4.30 y_{9} & -4.00 y_{10} & +  5.20 y_{11} & + 11.10 y_{12} & -0.50 y_{13} & -1.20 y_{14} & -4.10 y_{15}\\
 y_{8}   &  0.1 & -0.10 y_{4} & -0.90 y_{9} &   & -0.60 y_{11} & -0.30 y_{12} & -0.50 y_{13} & -0.40 y_{14} & -0.70 y_{15}\\
 y_{5}   &  0.0 & +  1.00 y_{4} & + 10.00 y_{9} & +  4.00 y_{10} & + 12.00 y_{11} & +  3.00 y_{12} & +  4.00 y_{13} & + 11.00 y_{14} & + 12.00 y_{15}\\
 y_{6}   &  1.7 & -0.70 y_{4} & +  3.70 y_{9} & +  9.00 y_{10} & -7.20 y_{11} & +  4.90 y_{12} & +  6.50 y_{13} & -9.80 y_{14} & +  1.10 y_{15}\\
 y_{7}   &  1.3 & -0.30 y_{4} & +  6.30 y_{9} & -10.00 y_{10} & -0.80 y_{11} & +  0.10 y_{12} & +  3.50 y_{13} & -10.20 y_{14} & -0.10 y_{15}\\
\hline
z    &  2.3 & -2.30 y_{4} & -41.70 y_{9} & -44.00 y_{10} & -8.80 y_{11} & -2.90 y_{12} & -12.50 y_{13} & -10.20 y_{14} & -45.10 y_{15}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  2.3 & +  0.30 x_{1} & +  0.70 x_{2} & -0.70 x_{3} & +  0.10 x_{8} & -1.00 x_{5} & +  0.70 x_{6} & +  0.30 x_{7}\\
 x_{9}   &  41.7 & -5.30 x_{1} & + 13.30 x_{2} & -4.30 x_{3} & +  0.90 x_{8} & -10.00 x_{5} & -3.70 x_{6} & -6.30 x_{7}\\
 x_{10}   &  44.0 & -7.00 x_{1} & -8.00 x_{2} & +  4.00 x_{3} &   & -4.00 x_{5} & -9.00 x_{6} & + 10.00 x_{7}\\
 x_{11}   &  8.8 & +  6.80 x_{1} & -0.80 x_{2} & -5.20 x_{3} & +  0.60 x_{8} & -12.00 x_{5} & +  7.20 x_{6} & +  0.80 x_{7}\\
 x_{12}   &  2.9 & +  3.90 x_{1} & +  0.10 x_{2} & -11.10 x_{3} & +  0.30 x_{8} & -3.00 x_{5} & -4.90 x_{6} & -0.10 x_{7}\\
 x_{13}   &  12.5 & + 10.50 x_{1} & +  7.50 x_{2} & +  0.50 x_{3} & +  0.50 x_{8} & -4.00 x_{5} & -6.50 x_{6} & -3.50 x_{7}\\
 x_{14}   &  10.2 & + 10.20 x_{1} & -2.20 x_{2} & +  1.20 x_{3} & +  0.40 x_{8} & -11.00 x_{5} & +  9.80 x_{6} & + 10.20 x_{7}\\
 x_{15}   &  45.1 & -4.90 x_{1} & +  1.90 x_{2} & +  4.10 x_{3} & +  0.70 x_{8} & -12.00 x_{5} & -1.10 x_{6} & +  0.10 x_{7}\\
\hline
z    &  -2.3 & -1.30 x_{1} & -1.70 x_{2} & -0.30 x_{3} & -0.10 x_{8} &   & -1.70 x_{6} & -1.30 x_{7}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  2.3 & +  0.30 x_{1} & +  0.70 x_{2} & -0.70 x_{3} & +  0.10 x_{8} & -1.00 x_{5} & +  0.70 x_{6} & +  0.30 x_{7}\\
 x_{9}   &  41.7 & -5.30 x_{1} & + 13.30 x_{2} & -4.30 x_{3} & +  0.90 x_{8} & -10.00 x_{5} & -3.70 x_{6} & -6.30 x_{7}\\
 x_{10}   &  44.0 & -7.00 x_{1} & -8.00 x_{2} & +  4.00 x_{3} &   & -4.00 x_{5} & -9.00 x_{6} & + 10.00 x_{7}\\
 x_{11}   &  8.8 & +  6.80 x_{1} & -0.80 x_{2} & -5.20 x_{3} & +  0.60 x_{8} & -12.00 x_{5} & +  7.20 x_{6} & +  0.80 x_{7}\\
 x_{12}   &  2.9 & +  3.90 x_{1} & +  0.10 x_{2} & -11.10 x_{3} & +  0.30 x_{8} & -3.00 x_{5} & -4.90 x_{6} & -0.10 x_{7}\\
 x_{13}   &  12.5 & + 10.50 x_{1} & +  7.50 x_{2} & +  0.50 x_{3} & +  0.50 x_{8} & -4.00 x_{5} & -6.50 x_{6} & -3.50 x_{7}\\
 x_{14}   &  10.2 & + 10.20 x_{1} & -2.20 x_{2} & +  1.20 x_{3} & +  0.40 x_{8} & -11.00 x_{5} & +  9.80 x_{6} & + 10.20 x_{7}\\
 x_{15}   &  45.1 & -4.90 x_{1} & +  1.90 x_{2} & +  4.10 x_{3} & +  0.70 x_{8} & -12.00 x_{5} & -1.10 x_{6} & +  0.10 x_{7}\\
\hline
z    &  2.3 & +  1.30 x_{1} & +  3.70 x_{2} & -0.70 x_{3} & +  0.10 x_{8} & -4.00 x_{5} & +  0.70 x_{6} & +  2.30 x_{7}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  2.3 & +  0.30 x_{1} & +  0.70 x_{2} & -0.70 x_{3} & +  0.10 x_{8} & -1.00 x_{5} & +  0.70 x_{6} & +  0.30 x_{7}\\
 x_{9}   &  41.7 & -5.30 x_{1} & + 13.30 x_{2} & -4.30 x_{3} & +  0.90 x_{8} & -10.00 x_{5} & -3.70 x_{6} & -6.30 x_{7}\\
 x_{10}   &  44.0 & -7.00 x_{1} & -8.00 x_{2} & +  4.00 x_{3} &   & -4.00 x_{5} & -9.00 x_{6} & + 10.00 x_{7}\\
 x_{11}   &  8.8 & +  6.80 x_{1} & -0.80 x_{2} & -5.20 x_{3} & +  0.60 x_{8} & -12.00 x_{5} & +  7.20 x_{6} & +  0.80 x_{7}\\
 x_{12}   &  2.9 & +  3.90 x_{1} & +  0.10 x_{2} & -11.10 x_{3} & +  0.30 x_{8} & -3.00 x_{5} & -4.90 x_{6} & -0.10 x_{7}\\
 x_{13}   &  12.5 & + 10.50 x_{1} & +  7.50 x_{2} & +  0.50 x_{3} & +  0.50 x_{8} & -4.00 x_{5} & -6.50 x_{6} & -3.50 x_{7}\\
 x_{14}   &  10.2 & + 10.20 x_{1} & -2.20 x_{2} & +  1.20 x_{3} & +  0.40 x_{8} & -11.00 x_{5} & +  9.80 x_{6} & + 10.20 x_{7}\\
 x_{15}   &  45.1 & -4.90 x_{1} & +  1.90 x_{2} & +  4.10 x_{3} & +  0.70 x_{8} & -12.00 x_{5} & -1.10 x_{6} & +  0.10 x_{7}\\
\hline
z    &  2.3 & +  1.30 x_{1} & +  3.70 x_{2} & -0.70 x_{3} & +  0.10 x_{8} & -4.00 x_{5} & +  0.70 x_{6} & +  2.30 x_{7}\\
\end{array}\]


 $ x_{1} $ enters and $ x_{10} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  4.18571428571 & -0.04 x_{10} & +  0.36 x_{2} & -0.53 x_{3} & +  0.10 x_{8} & -1.17 x_{5} & +  0.31 x_{6} & +  0.73 x_{7}\\
 x_{9}   &  8.38571428571 & +  0.76 x_{10} & + 19.36 x_{2} & -7.33 x_{3} & +  0.90 x_{8} & -6.97 x_{5} & +  3.11 x_{6} & -13.87 x_{7}\\
 x_{1}   &  6.28571428571 & -0.14 x_{10} & -1.14 x_{2} & +  0.57 x_{3} &   & -0.57 x_{5} & -1.29 x_{6} & +  1.43 x_{7}\\
 x_{11}   &  51.5428571429 & -0.97 x_{10} & -8.57 x_{2} & -1.31 x_{3} & +  0.60 x_{8} & -15.89 x_{5} & -1.54 x_{6} & + 10.51 x_{7}\\
 x_{12}   &  27.4142857143 & -0.56 x_{10} & -4.36 x_{2} & -8.87 x_{3} & +  0.30 x_{8} & -5.23 x_{5} & -9.91 x_{6} & +  5.47 x_{7}\\
 x_{13}   &  78.5 & -1.50 x_{10} & -4.50 x_{2} & +  6.50 x_{3} & +  0.50 x_{8} & -10.00 x_{5} & -20.00 x_{6} & + 11.50 x_{7}\\
 x_{14}   &  74.3142857143 & -1.46 x_{10} & -13.86 x_{2} & +  7.03 x_{3} & +  0.40 x_{8} & -16.83 x_{5} & -3.31 x_{6} & + 24.77 x_{7}\\
 x_{15}   &  14.3 & +  0.70 x_{10} & +  7.50 x_{2} & +  1.30 x_{3} & +  0.70 x_{8} & -9.20 x_{5} & +  5.20 x_{6} & -6.90 x_{7}\\
\hline
z    &  10.4714285714 & -0.19 x_{10} & +  2.21 x_{2} & +  0.04 x_{3} & +  0.10 x_{8} & -4.74 x_{5} & -0.97 x_{6} & +  4.16 x_{7}\\
\end{array}\]


 $ x_{2} $ enters and $ x_{14} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  6.10103092784 & -0.08 x_{10} & -0.03 x_{14} & -0.35 x_{3} & +  0.11 x_{8} & -1.61 x_{5} & +  0.23 x_{6} & +  1.37 x_{7}\\
 x_{9}   &  112.195876289 & -1.28 x_{10} & -1.40 x_{14} & +  2.49 x_{3} & +  1.46 x_{8} & -30.48 x_{5} & -1.52 x_{6} & + 20.73 x_{7}\\
 x_{1}   &  0.156701030928 & -0.02 x_{10} & +  0.08 x_{14} & -0.01 x_{3} & -0.03 x_{8} & +  0.82 x_{5} & -1.01 x_{6} & -0.61 x_{7}\\
 x_{11}   &  5.57525773196 & -0.07 x_{10} & +  0.62 x_{14} & -5.66 x_{3} & +  0.35 x_{8} & -5.48 x_{5} & +  0.51 x_{6} & -4.81 x_{7}\\
 x_{12}   &  4.04742268041 & -0.10 x_{10} & +  0.31 x_{14} & -11.08 x_{3} & +  0.17 x_{8} & +  0.06 x_{5} & -8.87 x_{6} & -2.32 x_{7}\\
 x_{13}   &  54.3670103093 & -1.03 x_{10} & +  0.32 x_{14} & +  4.22 x_{3} & +  0.37 x_{8} & -4.54 x_{5} & -18.92 x_{6} & +  3.46 x_{7}\\
 x_{2}   &  5.36288659794 & -0.11 x_{10} & -0.07 x_{14} & +  0.51 x_{3} & +  0.03 x_{8} & -1.21 x_{5} & -0.24 x_{6} & +  1.79 x_{7}\\
 x_{15}   &  54.5216494845 & -0.09 x_{10} & -0.54 x_{14} & +  5.10 x_{3} & +  0.92 x_{8} & -18.31 x_{5} & +  3.41 x_{6} & +  6.51 x_{7}\\
\hline
z    &  22.3463917526 & -0.42 x_{10} & -0.16 x_{14} & +  1.17 x_{3} & +  0.16 x_{8} & -7.43 x_{5} & -1.50 x_{6} & +  8.12 x_{7}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{12} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  5.97413712904 & -0.08 x_{10} & -0.04 x_{14} & +  0.03 x_{12} & +  0.10 x_{8} & -1.61 x_{5} & +  0.51 x_{6} & +  1.44 x_{7}\\
 x_{9}   &  113.10521909 & -1.30 x_{10} & -1.33 x_{14} & -0.22 x_{12} & +  1.50 x_{8} & -30.47 x_{5} & -3.51 x_{6} & + 20.21 x_{7}\\
 x_{1}   &  0.153688715229 & -0.02 x_{10} & +  0.08 x_{14} & +  0.00 x_{12} & -0.03 x_{8} & +  0.82 x_{5} & -1.01 x_{6} & -0.61 x_{7}\\
 x_{11}   &  3.50730300493 & -0.02 x_{10} & +  0.46 x_{14} & +  0.51 x_{12} & +  0.26 x_{8} & -5.51 x_{5} & +  5.04 x_{6} & -3.62 x_{7}\\
 x_{3}   &  0.365243278445 & -0.01 x_{10} & +  0.03 x_{14} & -0.09 x_{12} & +  0.02 x_{8} & +  0.01 x_{5} & -0.80 x_{6} & -0.21 x_{7}\\
 x_{13}   &  55.9074332496 & -1.06 x_{10} & +  0.44 x_{14} & -0.38 x_{12} & +  0.44 x_{8} & -4.51 x_{5} & -22.30 x_{6} & +  2.57 x_{7}\\
 x_{2}   &  5.5481440134 & -0.11 x_{10} & -0.06 x_{14} & -0.05 x_{12} & +  0.04 x_{8} & -1.21 x_{5} & -0.65 x_{6} & +  1.68 x_{7}\\
 x_{15}   &  56.3858963625 & -0.13 x_{10} & -0.40 x_{14} & -0.46 x_{12} & +  1.00 x_{8} & -18.28 x_{5} & -0.68 x_{6} & +  5.44 x_{7}\\
\hline
z    &  22.7722578845 & -0.43 x_{10} & -0.13 x_{14} & -0.11 x_{12} & +  0.18 x_{8} & -7.43 x_{5} & -2.43 x_{6} & +  7.87 x_{7}\\
\end{array}\]


 $ x_{7} $ enters and $ x_{1} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  6.33525660492 & -0.13 x_{10} & +  0.16 x_{14} & +  0.03 x_{12} & +  0.03 x_{8} & +  0.31 x_{5} & -1.86 x_{6} & -2.35 x_{1}\\
 x_{9}   &  118.17491649 & -2.05 x_{10} & +  1.39 x_{14} & -0.20 x_{12} & +  0.41 x_{8} & -3.53 x_{5} & -36.69 x_{6} & -32.99 x_{1}\\
 x_{7}   &  0.250835104768 & -0.04 x_{10} & +  0.13 x_{14} & +  0.00 x_{12} & -0.05 x_{8} & +  1.33 x_{5} & -1.64 x_{6} & -1.63 x_{1}\\
 x_{11}   &  2.59823868813 & +  0.11 x_{10} & -0.03 x_{14} & +  0.51 x_{12} & +  0.46 x_{8} & -10.34 x_{5} & + 10.99 x_{6} & +  5.91 x_{1}\\
 x_{3}   &  0.312784694807 & -0.00 x_{10} & +  0.00 x_{14} & -0.09 x_{12} & +  0.03 x_{8} & -0.27 x_{5} & -0.46 x_{6} & +  0.34 x_{1}\\
 x_{13}   &  56.5529911934 & -1.16 x_{10} & +  0.79 x_{14} & -0.38 x_{12} & +  0.30 x_{8} & -1.08 x_{5} & -26.53 x_{6} & -4.20 x_{1}\\
 x_{2}   &  5.96993622836 & -0.17 x_{10} & +  0.17 x_{14} & -0.04 x_{12} & -0.05 x_{8} & +  1.03 x_{5} & -3.41 x_{6} & -2.74 x_{1}\\
 x_{15}   &  57.7503795931 & -0.33 x_{10} & +  0.33 x_{14} & -0.45 x_{12} & +  0.70 x_{8} & -11.03 x_{5} & -9.61 x_{6} & -8.88 x_{1}\\
\hline
z    &  24.7467354995 & -0.72 x_{10} & +  0.93 x_{14} & -0.10 x_{12} & -0.24 x_{8} & +  3.06 x_{5} & -15.36 x_{6} & -12.85 x_{1}\\
\end{array}\]


 $ x_{5} $ enters and $ x_{11} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  6.4134892192 & -0.13 x_{10} & +  0.16 x_{14} & +  0.05 x_{12} & +  0.04 x_{8} & -0.03 x_{11} & -1.53 x_{6} & -2.17 x_{1}\\
 x_{9}   &  117.286880912 & -2.09 x_{10} & +  1.40 x_{14} & -0.37 x_{12} & +  0.25 x_{8} & +  0.34 x_{11} & -40.44 x_{6} & -35.01 x_{1}\\
 x_{7}   &  0.585747018389 & -0.02 x_{10} & +  0.13 x_{14} & +  0.07 x_{12} & +  0.01 x_{8} & -0.13 x_{11} & -0.22 x_{6} & -0.87 x_{1}\\
 x_{5}   &  0.251336584219 & +  0.01 x_{10} & -0.00 x_{14} & +  0.05 x_{12} & +  0.04 x_{8} & -0.10 x_{11} & +  1.06 x_{6} & +  0.57 x_{1}\\
 x_{3}   &  0.244168967746 & -0.00 x_{10} & +  0.00 x_{14} & -0.10 x_{12} & +  0.01 x_{8} & +  0.03 x_{11} & -0.75 x_{6} & +  0.19 x_{1}\\
 x_{13}   &  56.2811233183 & -1.17 x_{10} & +  0.79 x_{14} & -0.43 x_{12} & +  0.25 x_{8} & +  0.10 x_{11} & -27.67 x_{6} & -4.82 x_{1}\\
 x_{2}   &  6.22859996475 & -0.16 x_{10} & +  0.17 x_{14} & +  0.01 x_{12} & -0.01 x_{8} & -0.10 x_{11} & -2.31 x_{6} & -2.16 x_{1}\\
 x_{15}   &  54.977968392 & -0.46 x_{10} & +  0.36 x_{14} & -0.99 x_{12} & +  0.21 x_{8} & +  1.07 x_{11} & -21.34 x_{6} & -15.19 x_{1}\\
\hline
z    &  25.5167733976 & -0.69 x_{10} & +  0.92 x_{14} & +  0.05 x_{12} & -0.11 x_{8} & -0.30 x_{11} & -12.10 x_{6} & -11.09 x_{1}\\
\end{array}\]


 $ x_{12} $ enters and $ x_{3} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  6.52714932127 & -0.13 x_{10} & +  0.16 x_{14} & -0.47 x_{3} & +  0.05 x_{8} & -0.02 x_{11} & -1.87 x_{6} & -2.09 x_{1}\\
 x_{9}   &  116.409502262 & -2.07 x_{10} & +  1.39 x_{14} & +  3.59 x_{3} & +  0.19 x_{8} & +  0.25 x_{11} & -37.76 x_{6} & -35.67 x_{1}\\
 x_{7}   &  0.742081447964 & -0.02 x_{10} & +  0.13 x_{14} & -0.64 x_{3} & +  0.01 x_{8} & -0.11 x_{11} & -0.70 x_{6} & -0.75 x_{1}\\
 x_{5}   &  0.366515837104 & +  0.01 x_{10} & -0.00 x_{14} & -0.47 x_{3} & +  0.05 x_{8} & -0.08 x_{11} & +  0.71 x_{6} & +  0.66 x_{1}\\
 x_{12}   &  2.35067873303 & -0.04 x_{10} & +  0.01 x_{14} & -9.63 x_{3} & +  0.14 x_{8} & +  0.25 x_{11} & -7.20 x_{6} & +  1.78 x_{1}\\
 x_{13}   &  55.2692307692 & -1.15 x_{10} & +  0.79 x_{14} & +  4.14 x_{3} & +  0.19 x_{8} & -0.00 x_{11} & -24.58 x_{6} & -5.59 x_{1}\\
 x_{2}   &  6.2443438914 & -0.16 x_{10} & +  0.17 x_{14} & -0.06 x_{3} & -0.01 x_{8} & -0.10 x_{11} & -2.36 x_{6} & -2.14 x_{1}\\
 x_{15}   &  52.6402714932 & -0.42 x_{10} & +  0.35 x_{14} & +  9.57 x_{3} & +  0.07 x_{8} & +  0.81 x_{11} & -14.18 x_{6} & -16.96 x_{1}\\
\hline
z    &  25.6447963801 & -0.69 x_{10} & +  0.92 x_{14} & -0.52 x_{3} & -0.10 x_{8} & -0.28 x_{11} & -12.49 x_{6} & -11.00 x_{1}\\
\end{array}\]


 $ x_{14} $ enters and $ x_{5} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  32.0 & +  0.50 x_{10} & -69.50 x_{5} & -33.25 x_{3} & +  3.62 x_{8} & -5.87 x_{11} & + 47.50 x_{6} & + 43.75 x_{1}\\
 x_{9}   &  342.0 & +  3.50 x_{10} & -615.50 x_{5} & -286.75 x_{3} & + 31.87 x_{8} & -51.62 x_{11} & + 399.50 x_{6} & + 370.25 x_{1}\\
 x_{7}   &  22.0 & +  0.50 x_{10} & -58.00 x_{5} & -28.00 x_{3} & +  3.00 x_{8} & -5.00 x_{11} & + 40.50 x_{6} & + 37.50 x_{1}\\
 x_{14}   &  162.0 & +  4.00 x_{10} & -442.00 x_{5} & -208.50 x_{3} & + 22.75 x_{8} & -37.25 x_{11} & + 314.00 x_{6} & + 291.50 x_{1}\\
 x_{12}   &  4.0 & +  0.00 x_{10} & -4.50 x_{5} & -11.75 x_{3} & +  0.38 x_{8} & -0.13 x_{11} & -4.00 x_{6} & +  4.75 x_{1}\\
 x_{13}   &  183.0 & +  2.00 x_{10} & -348.50 x_{5} & -160.25 x_{3} & + 18.12 x_{8} & -29.37 x_{11} & + 223.00 x_{6} & + 224.25 x_{1}\\
 x_{2}   &  33.0 & +  0.50 x_{10} & -73.00 x_{5} & -34.50 x_{3} & +  3.75 x_{8} & -6.25 x_{11} & + 49.50 x_{6} & + 46.00 x_{1}\\
 x_{15}   &  110.0 & +  1.00 x_{10} & -156.50 x_{5} & -64.25 x_{3} & +  8.12 x_{8} & -12.37 x_{11} & + 97.00 x_{6} & + 86.25 x_{1}\\
\hline
z    &  175.0 & +  3.00 x_{10} & -407.50 x_{5} & -192.75 x_{3} & + 20.87 x_{8} & -34.62 x_{11} & + 277.00 x_{6} & + 257.75 x_{1}\\
\end{array}\]


 $ x_{1} $ enters and Unbounded Dictionary!


 $ x_{1} $ enters and Unbounded Dictionary!
\end{document}
